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Full-Text Articles in Physical Sciences and Mathematics
On Similarity Classes Of Well-Rounded Sublattices Of Z², Lenny Fukshansky
On Similarity Classes Of Well-Rounded Sublattices Of Z², Lenny Fukshansky
CMC Faculty Publications and Research
A lattice is called well-rounded if its minimal vectors span the corresponding Euclidean space. In this paper we study the similarity classes of well-rounded sublattices of Z2. We relate the set of all such similarity classes to a subset of primitive Pythagorean triples, and prove that it has the structure of a non-commutative infinitely generated monoid. We discuss the structure of a given similarity class, and define a zeta function corresponding to each similarity class. We relate it to Dedekind zeta of Z[i], and investigate the growth of some related Dirichlet series, which reflect on …
Sphere Packing, Lattices, And Epstein Zeta Function, Lenny Fukshansky
Sphere Packing, Lattices, And Epstein Zeta Function, Lenny Fukshansky
CMC Faculty Publications and Research
The sphere packing problem in dimension N asks for an arrangement of non-overlapping spheres of equal radius which occupies the largest possible proportion of the corresponding Euclidean space. This problem has a long and fascinating history. In 1611 Johannes Kepler conjectured that the best possible packing in dimension 3 is obtained by a face centered cubic and hexagonal arrangements of spheres. A proof of this legendary conjecture has finally been published in 2005 by Thomas Hales. The analogous problem in dimension 2 has been solved by Laszlo Fejes Toth in 1940, and this really is the extent of our current …
On Distribution Of Integral Well-Rounded Lattices In Dimension Two, Lenny Fukshansky
On Distribution Of Integral Well-Rounded Lattices In Dimension Two, Lenny Fukshansky
CMC Faculty Publications and Research
Lecture given at the Illinois Number Theory Fest, May 2007.